Optimality Study of Logic Synthesis for LUT-Based Fpgas Jason Cong and Kirill Minkovich

230 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 26, NO. 2, FEBRUARY 2007 Optimality Study of Logic Synthesis for LUT-Based FPGAs Jason Cong and Kirill Minkovich Abstract—Field-programmable gate-array (FPGA) logic synthesis and technology mapping have been studied extensively over the past 15 years. However, progress within the last few years has slowed considerably (with some notable exceptions). It seems natural to then question whether the current logic-synthesis and technology-mapping algorithms for FPGA designs are producing near-optimal solutions. Although there are many empirical studies that compare different FPGA synthesis/mapping algorithms, little is known about how far these algorithms are from the optimal (recall that both logic-optimization and technology- mapping problems are NP-hard, if we consider area optimization in addition to delay/depth optimization). In this paper, we present a novel method for constructing arbitrarily large circuits that have known optimal solutions after technology mapping. Using these circuits and their derivatives (called Logic synthesis Exam- ples with Known Optimal (LEKO) and Logic synthesis Exam- ples with Known Upper bounds (LEKU), respectively), we show that although leading FPGA technology-mapping algorithms can produce close to optimal solutions, the results from the entire logic-synthesis flow (logic optimization + mapping) are far from optimal. The LEKU circuits were constructed to show where the logic synthesis flow can be improved, while the LEKO circuits specifically deal with the performance of the technology mapping. The best industrial and academic FPGA synthesis flows Fig. 1. Possible area-minimal mapping solutions. (a) Original circuit. (b) Mapping solution without logic optimization. (c) Mapping solution with are around 70 times larger in terms of area on average and, in logic optimization. some cases, as much as 500 times larger on LEKU examples. These results clearly indicate that there is much room for further research and improvement in FPGA synthesis. capable of implementing any K-input one-output Boolean function. Index Terms—Circuit optimization, circuit synthesis, design automation, field-programmable gate arrays (FPGAs), optimiza- Given a register transfer level (RTL) design, the typical tion methods. FPGA synthesis process consists of RTL elaboration, logic synthesis, and the physical design (layout synthesis) [11]. In this paper, we will focus on logic synthesis, which can be broken I. INTRODUCTION down into two main steps: logic optimization and technology IELD-PROGRAMMABLE gate arrays (FPGAs) have mapping. Logic optimization transforms the current gate-level F been gaining momentum as an alternative to application- network into an equivalent gate-level network more suitable for specific integrated circuits (ASICs). FPGAs consist of program- technology mapping. Technology mapping transforms the gate- mable logic, input-output (I/O), and routing elements, which level network into a network of programmable cells (in our can be programmed and reprogrammed in the field to customize case, these cells are LUTs) by covering the network with these an FPGA, enabling it to implement a given application in a cells. Several algorithms perform logic optimization during matter of seconds or milliseconds. The most common type technology mapping. As an example, Fig. 1 shows the differ- of programmable-logic element used in an FPGA is called a ence between mapping algorithms that use logic optimization K-LUT, which is a K-input one-output lookup table (LUT), and those that do not. By examining the logic function of f,we can see that it just takes the logical AND of all its inputs; thus, by manipulating the circuit, we can reduce the mapping solution Manuscript received March 16, 2006; revised July 17, 2006 and by one 4-LUT. Since the size of the circuit will be directly September 13, 2006. This work was supported in part by the National Science proportional to the price of an FPGA that can implement Foundation under Grant CCF-0306682 and Grant CCF 0430077 and in part by it, the logic-synthesis step will play an integral role in the Altera Corporation, Magma Design Automation Inc., and Xilinx Inc. under the California MICRO Program. This paper was recommended by Associate Editor design flow. K. Bazargan. As FPGA technology gained popularity throughout the The authors are with the Computer Science Department, University of 1990s, a large amount of work was published that dealt with California, Los Angeles, CA 90095 USA (e-mail: [email protected]; cory_m@ cs.ucla.edu). logic synthesis and/or technology mapping of FPGAs, includ- Digital Object Identifier 10.1109/TCAD.2006.887922 ing Chortle-crf [20], XMap [24], TechMap [31], DAG-map [9], 0278-0070/$25.00 © 2007 IEEE CONG AND MINKOVICH: OPTIMALITY STUDY OF LOGIC SYNTHESIS FOR LUT-BASED FPGAs 231 FlowMap [13], Zmap [16], Cutmap [12], BoolMap [27], and this approach (SAT-based optimal logic optimization is applied many others. More comprehensive surveys of FPGA synthe- to logic cones of up to ten inputs). sis and mapping algorithms are available from [8] and [11]. In this paper, we present a novel method for constructing These mapping algorithms employ many different techniques arbitrarily large circuits that have known optimal solutions after to achieve their solutions, including dynamic programming, bin technology mapping, or known upper bound solutions after packing, BDD-based logic simplification, and cut enumeration, logic optimization and technology mapping for LUT-based just to name a few. Some of these algorithms focused on delay FPGAs. Using these circuits [called Logic synthesis Examples minimization [14], [20], [24], [29], [31], [35], while others with Known Optimal (LEKO) and Logic synthesis Examples focused on area minimization possible under delay or depth with Known Upper bounds (LEKU)], we show that although constraints [9], [10], [12], [13], [16], [27], [31], [36]. All of leading FPGA technology-mapping algorithms can produce these algorithms were developed over a ten-year period in the close to optimal solutions, the results from the entire logic- 1990s but, after this influx, the amount of new published work synthesis flow (logic optimization + mapping) are far from op- began to decrease steadily with only a few novel algorithms timal. The best industrial and academic FPGA synthesis flows emerging in the past few years—such as IMAP [3], Hermes are around 70 times larger in terms of area on average and, in [19], DAOmap [7], and the ABC mapper [1], [38]. To many some cases, as much as 500 times larger on LEKU examples. people, this signaled that FPGA synthesis algorithms had prob- These results clearly indicate that there is much room for further ably hit a plateau. It is natural to then question whether the research and improvement in FPGA synthesis. current logic-synthesis and technology-mapping algorithms for FPGA designs are producing near-optimal solutions. Although II. CONSTRUCTION OF BENCHMARKS there are many empirical studies that compare different FPGA synthesis/mapping algorithms, little is known about how far A. Construction of LEKO Examples these algorithms are from the optimal (recall that both logic- We present an algorithm for constructing a network Gn (with optimization and technology-mapping problems are NP-hard n inputs and outputs) of an arbitrarily large size that has a if we consider area optimization in addition to delay/depth known optimal technology mapping solution. Gn is constructed optimization). in a special way by replicating a small circuit with a known In fact, a similar question was raised a few years ago when optimal mapping solution into a circuit of any size that also has placement research slowed down. However, using a set of a known optimal mapping solution. These circuits are called cleverly constructed examples, called placement examples with LEKO examples. The building block of our construction is a known optimal (PEKO), the study in [6] showed surprising re- “core graph” named Cn, with the following properties. sults: Wirelengths produced by state-of-the-art placement tools n n at that time were 1.66–2.53 times the optimal solutions in the 1) It has inputs and outputs. n worst cases. These results generated a renewed interest in place- 2) Every output is a function of all inputs. C ment research; within two to three years, a large body of papers 3) Each internal node of n has exactly two inputs. was published on placement optimality studies (e.g., [15], [17], 4) There exists an optimal (in terms of area/depth) mapping C M [18], and [23]), as well as novel placement algorithms (e.g., of n into a 4-LUT mapping solution, denoted as n, M [4], [5], [22], [30], and [34]). Within three years, the optimality such that n only has 4-LUTs (no 3-LUTs or 2-LUTs). C M gap on the PEKO examples was reduced to roughly 20% on For the 5 showninFig.2, 5 has exactly seven 4-LUTs. average [4]. The actual improvement on the IBM (ISPD04) This general method can be used to create core graphs benchmarks was 24% by the mPL placer [33]. This leads one of arbitrary size, although in this paper, we will only present to believe that the improvement on the artificially constructed how C5 and C6 are constructed. These core graphs target the PEKO examples correlates to some extent the improvement 4-LUT architecture, because it is the simplest of those currently on the “real” (more realistic) examples. This might be due to in use. But, the ideas behind our construction can be extended to the fact that a small suboptimality of an algorithm on the real Altera’s Adaptive Logic Module or any sized LUT architecture benchmarks often gets magnified into a significant optimality by only adapting the fourth property to reflect an optimal gap on the PEKO benchmarks.

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